2 Mechanical Equilibrium 2 Mechanical Equilibrium Things that are in balance with one another illustrate equilibrium. Things in mechanical equilibrium are stable, without changes of motion. The rocks are in mechanical equilibrium. An object in mechanical An unbalanced external force would be equilibrium is stable, without needed to changgge their resting state. changes in motion.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.1 Force 2.1 Force Net Force A force is a push or a pull. A force of some kind is always required to change the state of motion of an object. A force is needed to change an object’s The combination of all forces acting on an object is state of motion . called the net force. The net force on an object changes its motion. The scientific unit of force is the newton, abbreviated N.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.1 Force 2.1 Force Net Force Net Force The net force depends on The net force depends on the magnitudes and the magnitudes and directions of the applied directions of the applied forces. forces.
1 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.1 Force 2.1 Force Net Force Net Force The net force depends on The net force depends on the magnitudes and the magnitudes and directions of the applied directions of the applied forces. forces.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.1 Force 2.1 Force Net Force Net Force The net force depends on The net force depends on the magnitudes and the magnitudes and directions of the applied directions of the applied forces. forces.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.1 Force 2.1 Force Net Force Tension and Weight When the girl holds the A stretched spring is under a “stretching force” rock with as much force called tension. upward as gravity pulls Pounds and newtons are units of weight, which downward, the net force are units of force. on the rock is zero .
2 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.1 Force 2.1 Force Tension and Weight Tension and Weight The upward tension in the The upward tension in the string has the same magnitude string has the same magnitude as the weight of the bag, so the as the weight of the bag, so the net force on the bag is zero. net force on the bag is zero. The bag of sugar is attracted to The bag of sugar is attracted to Earth with a gravitational force Earth with a gravitational force of 2 pounds or 9 newtons. of 2 pounds or 9 newtons.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.1 Force 2.1 Force Tension and Weight Force Vectors There are two forces acting on the bag of sugar: A vector is an arrow that represents the magnitude and • tension force acting upward direction of a quantity. • weight acting downward A vector quantity needs both magnitude and direction for a The tw o f or ces on th e bag ar e equal an d opposi te. Th e complete description. Force is an example of a vector quantity. net force on the bag is zero, so it remains at rest. A scalar quantity can be described by magnitude only and has no direction. Time, area, and volume are scalar quantities.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.1 Force 2.1 Force Force Vectors Force Vectors This vector represents a force of 60 N to the right.
3 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.1 Force 2.2 Mechanical Equilibrium
You can express the equilibrium rule How can you change an object’s state mathematically as F =0= 0. of motion?
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.2 Mechanical Equilibrium 2.2 Mechanical Equilibrium Mechanical equilibrium is a state wherein no physical changes occur. Whenever the net force on an object is zero, the object is in mechanical equilibrium—this is known as the equilibrium rule. • The symbol stands for “the sum of.” • F stands f or “f or ces.” For a suspended object at rest, the forces acting upward on the object must be balanced by other forces acting downward. The vector sum equals zero.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.2 Mechanical Equilibrium 2.2 Mechanical Equilibrium The sum of the upward vectors equals the sum of the The sum of the upward vectors equals the sum of the downward vectors. F = 0, and the scaffold is in equilibrium. downward vectors. F = 0, and the scaffold is in equilibrium.
4 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.2 Mechanical Equilibrium 2.2 Mechanical Equilibrium The sum of the upward vectors equals the sum of the The sum of the upward vectors equals the sum of the downward vectors. F = 0, and the scaffold is in equilibrium. downward vectors. F = 0, and the scaffold is in equilibrium.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.2 Mechanical Equilibrium 2.2 Mechanical Equilibrium think! think! If the gymnast hangs with her weight evenly If the gymnast hangs with her weight evenly divided between the two rings, how would scale divided between the two rings, how would scale readings in both supporting ropes compare with readings in both supporting ropes compare with her weight? Suppose she hangs with slightly her weight? Suppose she hangs with slightly more of her weight supported by the left ring. more of her weight supported by the left ring. How would a scale on the right read? How would a scale on the right read?
Answer: In the first case, the reading on each scale will be half her weight. In the second case, when more of her weight is supported by the left ring, the reading on the right reduces to less than half her weight. The sum of the scale readings always equals her weight.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.2 Mechanical Equilibrium 2.3 Support Force
For an object at rest on a horizontal surface, the How can you express the equilibrium support force must equal the object’ s weight. rule mathematically?
5 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.3 Support Force 2.3 Support Force What forces act on a book lying at rest on a table? • One is the force due to gravity—the weight of the book. • There must be another force acting on it to produce a net force of zero—an upward force opposite to the force of gravity. The table pushes up on The u pw ard force that balances the w eight of an object on a the book with as much surface is called the support force. force as the downward weight of the book. A support force is often called the normal force.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.3 Support Force 2.3 Support Force The table supports the book with a support force— • The upward support force is positive and the downward the upward force that balances the weight of an object weight is negative. on a surface. • The two forces add mathematically to zero. A support force is often called the normal force. • Another way to say the net force on the book is zero is F = 0. The book lying on the table compresses atoms in the table and they squeeze upward on the book. The compressed atoms produce the support force.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.3 Support Force 2.3 Support Force The upward support The upward support force is as much as the force is as much as the downward pull of downward pull of gravity. gravity.
6 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.3 Support Force 2.3 Support Force think! think! What is the net force on a bathroom scale when a 110-pound What is the net force on a bathroom scale when a 110-pound person stands on it? person stands on it?
Answer: Zero–the scale is at rest. The scale reads the support ffforce, not the net force.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.3 Support Force 2.3 Support Force think! think! Suppose you stand on two bathroom scales with your weight Suppose you stand on two bathroom scales with your weight evenly distributed between the two scales. What is the reading evenly distributed between the two scales. What is the reading on each of the scales? What happens when you stand with on each of the scales? What happens when you stand with more of your weight on one foot than the other? more of your weight on one foot than the other?
Answer: In the first case, the reading on each scale is half your weight. In the second case, if you lean more on one scale than the other, more than half your weight will be read on that scale but less than half on the other. The total support force adds up to your weight.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.3 Support Force 2.4 Equilibrium for Moving Objects
Objects at rest are said to be in static equilibrium; For an object at rest on a horizontal surface, objects moving at constant speed in a straight-line what is the support force equal to? path are said to be in dynamic equilibrium.
7 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.4 Equilibrium for Moving Objects 2.4 Equilibrium for Moving Objects The state of rest is only one form of equilibrium. An object under the influence of only one force cannot An object moving at constant speed in a straight-line path is be in equilibrium. also in a state of equilibrium. Once in motion, if there is no net Only when there is no force at all, or when two or more force to change the state of motion, it is in equilibrium. forces combine to zero, can an object be in equilibrium.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.4 Equilibrium for Moving Objects 2.4 Equilibrium for Moving Objects When the push on the If the desk moves steadily at constant speed, without change desk is the same as the in its motion, it is in equilibrium. force of friction between • Friction is a contact force between objects that slide or the desk and the floor, tend to slide against each other. the net force is zero • In this case, F = 0 means that the force of friction is and the desk slides at equal in magnitude and opposite in direction to the an unchanging speed. pushing force.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.4 Equilibrium for Moving Objects 2.4 Equilibrium for Moving Objects think! think! An airplane flies horizontally at constant speed in a straight- An airplane flies horizontally at constant speed in a straight- line direction. Its state of motion is unchanging. In other line direction. Its state of motion is unchanging. In other words, it is in equilibrium. Two horizontal forces act on the words, it is in equilibrium. Two horizontal forces act on the plane. One is the thrust of the propeller that pulls it forward. plane. One is the thrust of the propeller that pulls it forward. The other is the force of air resistance (air friction) that acts in The other is the force of air resistance (air friction) that acts in the opposite direction. Which force is greater? the opposite direction. Which force is greater?
Answer: Neither, for both forces have the same strength. Call the thrust positive. Then the air resistance is negative. Since the plane is in equilibrium, the two forces combine to equal zero.
8 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.4 Equilibrium for Moving Objects 2.5 Vectors
To find the resultant of two vectors, construct a parallelogram wherein the two vectors are How are static and dynamic adjacent sides . The diagonal of the equilibrium different? parallelogram shows the resultant.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.5 Vectors 2.5 Vectors The sum of two or more vectors is called their resultant. a. The tension in the rope Combining vectors is quite simple when they are parallel: is 300 N, equal to • If they are in the same direction, they add. Nellie’s weight. • If they are in opposite directions, they subtract.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.5 Vectors 2.5 Vectors a. The tension in the rope The Parallelogram Rule is 300 N, equal to To find the resultant of nonparallel vectors, we use the Nellie’s weight. parallelogram rule. b. The tension in each rope Consider two vectors at right angles to each other, as shown is now 150 N, half of below. The constructed parallelogram in this special case is a Nellie’ s weight. In each rectangle. The diagonal is the resultant R. case, F = 0.
9 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.5 Vectors 2.5 Vectors The Parallelogram Rule Applying the Parallelogram Rule In the special case of two perpendicular vectors that are equal When Nellie is suspended at in magnitude, the parallelogram is a square. rest from the two non-vertical The resultant is times one of the vectors. ropes, is the rope tension greater or less than the For example, the resultant of two equal vectors of magnitude tension in two vertical ropes? 100 acting at a right angle to each other is 141.4. You need to use the parallelogram rule to determine the tension.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.5 Vectors 2.5 Vectors Applying the Parallelogram Rule Applying the Parallelogram Rule Notice how the tension vectors form a parallelogram in which Nellie’s weight is shown by the downward vertical vector. the resultant R is vertical. An equal and opposite vector is needed for equilibrium, shown by the dashed vector. Note that the dashed vector is the diagonal of the parallelogram defined by the dotted lines. Using the parallelogram rule, we find that the tension in each rope is more than half her weight.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.5 Vectors 2.5 Vectors Applying the Parallelogram Rule Applying the Parallelogram Rule As the angle between the ropes increases, tension increases so that the When the ropes supporting Nellie are at different angles to the vertical, the resultant (dashed-line vector) remains at 300 N upward, which is required tensions in the two ropes are unequal. to support 300-N Nellie. By the parallelogram rule, we see that the right rope bears most of the load and has the greater tension.
10 2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.5 Vectors 2.5 Vectors Applying the Parallelogram Rule think! You can safely hang from a clothesline hanging vertically, but Two sets of swings you will break the clothesline if it is strung horizontally. are shown at right. If the children on the swings are of equal weights, the ropes of which swing are more likely to break?
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.5 Vectors 2.5 Vectors think! think! Two sets of swings Consider what would happen if you suspended a 10-N object are shown at right. midway along a very tight, horizontally stretched guitar string. If the children on the Is it possible for the string to remain horizontal without a slight swings are of equal sag at the point of suspension? weights, the ropes of which swing are more likely to break?
Answer: The tension is greater in the ropes hanging at an angle. The angled ropes are more likely to break than the vertical ropes.
2 Mechanical Equilibrium 2 Mechanical Equilibrium 2.5 Vectors 2.5 Vectors think! Consider what would happen if you suspended a 10-N object midway along a very tight, horizontally stretched guitar string. Is it possible for the string to remain horizontal without a slight sag at the point of suspension? How can you find the resultant of two vectors? Answer: No way! If the 10-N load is to hang in equilibrium, there must be a supporting 10-N upward resultant. The tension in each half of the guitar string must form a parallelogram with a vertically upward 10-N resultant.
11 2 Mechanical Equilibrium 2 Mechanical Equilibrium Assessment Questions Assessment Questions 1. When you hold a rock in your hand at rest, 1. When you hold a rock in your hand at rest, the forces on the rock the forces on the rock a. are mainly due to gravity. a. are mainly due to gravity. b. are mainly due to the upward push of b. are mainly due to the upward push of your hand. your hand. c. cancel to zero. c. cancel to zero. d. d’ttdon’t act un less thkidthe rock is droppe d. d. d’ttdon’t act un less thkidthe rock is droppe d.
Answer: C
2 Mechanical Equilibrium 2 Mechanical Equilibrium Assessment Questions Assessment Questions 2. Burl and Paul have combined weights of 1300 N. The tensions in 2. Burl and Paul have combined weights of 1300 N. The tensions in the supporting ropes that support the scaffold they stand on add to the supporting ropes that support the scaffold they stand on add to 1700 N. The weight of the scaffold itself must be 1700 N. The weight of the scaffold itself must be a. 400 N. a. 400 N. b. 500 N. b. 500 N. c. 600 N. c. 600 N. d. 3000 N. d. 3000 N.
Answer: A
2 Mechanical Equilibrium 2 Mechanical Equilibrium Assessment Questions Assessment Questions 3. Harry gives his little sister a piggyback ride. Harry weighs 400 N and 3. Harry gives his little sister a piggyback ride. Harry weighs 400 N and his little sister weighs 200 N. The support force supplied by the floor his little sister weighs 200 N. The support force supplied by the floor must be must be a. 200 N. a. 200 N. b. 400 N. b. 400 N. c. 600 N. c. 600 N. d. more than 600 N. d. more than 600 N.
Answer: C
12 2 Mechanical Equilibrium 2 Mechanical Equilibrium Assessment Questions Assessment Questions 4. When a desk is horizontally pushed across a floor at a steady speed 4. When a desk is horizontally pushed across a floor at a steady speed in a straight-line direction, the amount of friction acting on the desk is in a straight-line direction, the amount of friction acting on the desk is a. less than the pushing force. a. less than the pushing force. b. equal to the pushing force. b. equal to the pushing force. c. greater than the pushing force. c. greater than the pushing force. d. dependent on the speed of the sliding crate. d. dependent on the speed of the sliding crate.
Answer: B
2 Mechanical Equilibrium 2 Mechanical Equilibrium Assessment Questions Assessment Questions 5. When Nellie hangs at rest by a pair of ropes, the tensions in the 5. When Nellie hangs at rest by a pair of ropes, the tensions in the ropes ropes a. always equal her weight. a. always equal her weight. b. always equal half her weight. b. always equal half her weight. c. depend on the angle of the ropes to the vertical. c. depend on the angle of the ropes to the vertical. d. are twice her weight. d. are twice her weight.
Answer: C
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